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%I #17 Mar 14 2024 04:53:07
%S 0,0,6,30,140,560,2058,7098,23472,75372,237182,735878,2260596,6896136,
%T 20933778,63325170,191089112,575626052,1731858246,5206059774,
%U 15640198620,46966732320,140996664986,423191320490,1269993390720
%N Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
%D J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).
%H J. Propp, <a href="http://faculty.uml.edu/jpropp/articles.html">Publications and Preprints</a>
%H J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="https://library.slmath.org/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (9,-30,42,-9,-39,40,-12).
%F a(n) = (3^(n+2)+(-1)^(n+2))/2-2^(n+2)-(n+2)(2^(n+1)-1)+(n+1)^2.
%F a(n) = Entry n+2 in row n of (Sequence to be added #1).
%F a(n) = A046717(n+2)-2^(n+2)-(n+2)(2^(n+1)-1)+(n+1)^2.
%F a(n) = 9*a(n-1)-30*a(n-2)+42*a(n-3)-9*a(n-4)-39*a(n-5)+40*a(n-6)-12*a(n-7). [_Harvey P. Dale_, Nov 27 2011]
%F G.f.: 2*x^2*(6*x^4-26*x^3+25*x^2-12*x+3)/((x-1)^3*(x+1)*(2*x-1)^2*(3*x-1)). [_Colin Barker_, Nov 22 2012]
%e a(3)=(3^5+(-1)^5)/2-2^5-5(2^4-1)+4^2=30.
%t Table[(3^(n+2)+(-1)^(n+2))/2-2^(n+2)-(n+2)(2^(n+1)-1)+(n+1)^2,{n,0,30}] (* or *) LinearRecurrence[{9,-30,42,-9,-39,40,-12},{0,0,6,30,140,560,2058},30] (* _Harvey P. Dale_, Nov 27 2011 *)
%Y Cf. A092437-A092443.
%K easy,nonn
%O 0,3
%A Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004
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